Certain Formulas for the Hankel–Clifford Hyper-Transforms

In recent years, multi-index analogs of Bessel functions have emerged as powerful tools across various branches of mathematics. In this work, we present two distinct group-theoretic proofs of a key identity involving the difference between the first and second Hankel–Clifford integral transforms in the multi-index setting. The first approach leverages the interplay between different bases within a single representation space, while the second highlights the relationship between two distinct representation spaces. Additionally, we derive an alternative formula that further characterizes the difference between these transforms, offering new insights with potential applications in both theoretical and applied contexts.